Bayesian network is a type of probabilistic graphical model based on Bayesian theorem, which is a directed acyclic graph containing nodes and directed edges. The directed edge represents the causality or association among nodes. The association strength is determined by the conditional probability table. Generally speaking, Bayesian theorem is that you can predict the probability of an event relying on the probability of other events related to the nature of the event when you can’t determine the probability of an event occurrence. As mathematicians say, the more events supporting an attribute occur, the greater the likelihood that it will occur [
19].
2.1. Selection of Network Node
There are many variables during aluminum reduction. It is necessary to select variables with obvious causality with heat balance state as nodes of Bayesian network. Through state discretization of 500 kA cell production data in a Chinese smelter according to the actual situation and variable characteristics, the specific nodes and state discretization are listed in
Table 1.
(1) “Heat_state”
This node indicates the heat balance state of the cell based on its superheat value.
(2) “Often_AE”
When the superheat decreases, alumina is not easily dissolved and more anode effects will occur, which can be acquired in the cell control system in real time.
(3) “Block”
When superheat decreases, undissolved alumina will accumulate near the feeding hole and cause blockage. If the blocked hole is not cleaned, the block is only counted once. After cleanup, if the block occurs on this hole again, another block event can be counted. An intelligent breaking and feeding system was applied to monitor the block state of each feeding hole, the method of which is in [
20].
(4) “Bath_level”
This node represents the height of the bath level.
(5) “Metal_level”
This node represents the height of the metal level.
(6) “Superheat”
“Superheat” is calculated based on electrolyte temperature and liquidus temperature. The empirical formula for liquidus temperature calculation is given in the literature [
21].
(7) “Heat_long”
Firstly, the current efficiency for a longer period of time (the latest week in general) is approximately current efficiency
η according to tapping amount, so electrochemical reaction consumption and energy for heating materials [
22] is
WR = (0.48 + 1.644 ×
CE/100) ×
I (kW).
The energy input is equal to voltage times current: Win = U × I (kW).
In the above two formulas, U refers to cell voltage and I refers to potline current, both of which can be acquired from the cell control system in real time.
In accordance with the law of energy balance, the heat loss from a cell surface equals Win − WR, which is also the heat for maintaining cell temperature and electrolyte superheat. This heat value is estimated based on average current efficiency for a longer period of time, so it is called “Heat_long”. For easy understanding, voltage is used to express “Heat_long”, i.e., U − (0.48 + 1.644 × CE/100).
(8) “MHD” (Magnetohydrodynamic)
The MHD fluctuation index can be acquired from anode current monitoring system. The higher the index, the more fluctuations occur between the molten metal and electrolyte interface.
(9) “CVD”
Both cathode block quality and proper heating-up process can influence cathode voltage drop (CVD), but those influence are long and stable. The value of “CVD” must be determined based on the actual conditions of each cell. After verifying that there is no hard precipitation on the cathode and that the heat balance state remains normal, CVD (V0) is measured as baseline.
(10) “Spike”
All anode currents were measured by an online anode current measuring system [
23,
24]. Anode current can obviously reflect the anode spike. As for the anodes with higher current and lower noise, the anode spike can be inspected on the monitor by operators.
(11) “Temp_change”
This variable represents the variation trend of bath temperature.
The temperature measurement precision of thermocouple is 0.1 °C. The three latest values of electrolyte temperatures (in chronological order T2, T1, and T0; T0 represents the temperature measured that day) are used to conduct trend analysis.
(12) “Heat_xstate”
The variable is not an observed variable but a known variable, which is a historical value of previous “Heat_state”. It is based on the manual measurement of superheat. Its classification is consistent with “Heat_state”.
(13) “Wall_temp”
This node refers to the temperature of the side wall surface.
(14) “Sludge”
The sludge on the cathode surface can be judged by operators.
(15) “Heat_present”
“Heat_present” is jointly determined by “Spike” and “Heat_long”. “Heat_present” refers to the current heat loss considering anode spikes, which is the correction of “Heat long”, so its state classification is consistent with “Heat long”.
2.2. Analysis of Causality among Nodes
2.2.2. Variables Affected by “Heat_State”
(1) “Often AE”
As mentioned above, some alumina cannot be dissolved when the superheat is low, and it can easily lead to the occurrence of anode effects.
In addition, the variable “Bath level” influences “Often_AE”. When the “Bath_level” is low, i.e., insufficient electrolyte to dissolve the amount of alumina, this results in more anode effects.
As the electrolyte ratio (concentration ratio of NaF/AlF
3) decreases, the saturated solubility of alumina decreases, and the alumina dissolution rate decreases as well. The current alumina concentration can generally be controlled below 3%, and the appropriately low electrolyte ratio will not cause an anode effect [
25]. In addition, many aluminum smelters target achieving a low electrolyte ratio. The anode effect will occur frequently (i.e., 1 AE/pot∙day) only when the electrolyte ratio is very low. In this paper, it is assumed that the electrolyte ratio can be controlled within a reasonable range.
(2) “Superheat”
Although “Superheat” has an error in calculation, unable to exactly substitute the actual superheat, it has a certain relation with the actual superheat.
(3) “Wall_temp”
When the cell maintains high superheat, “Wall temp” will become very high.
(4) “Block”
When the cell maintains low superheat, alumina is not easily dissolved and the feeding hole is easily blocked.
(5) “Sludge”
When the cell retains low superheat for a long time, the alumina easily precipitates on the cathode surface and hard precipitation forms and, therefore, “Sludge” becomes worse.
(6) “Temp_change”
“Heat_xstate” and “Heat state” have joint influence on “Temp_change”. If the cell changes from low to high superheat, “Temp_change” presents an upward trend (i.e., state of “increase”); if the cell changes from high to low superheat, “Temp_change” presents a downward trend (i.e., state of “decrease”).
In addition, “Heat_state” can also influence “Bath level”, but such influence is ignored since the operators often tap or add electrolyte, giving rise to great disturbance on the “Bath level”.